Those are different integration techniques that are useful in different situations, depending upon what the form of your function being integrated is. I would recommend going back to you calculus text and looking up in what context each technique was introduced:

A summary would be:

1. Substitution: obviously useful when your integrand (the function being integrated) is actually the derivative of composition of two functions i.e. it has the form of something that has been differentiated using the chain rule.

2. By parts: useful when your integrand looks like one of the terms of something that has been differentiated using the product rule.

3. Trigonometry. This should be obvious! If your integrand has trigonometric functions, then it is applicable, otherwise it isn't.

4. Partial fractions: Umm...again fairly self explanatory. If I recall correctly, this is useful if your integrand can be decomposed using partial fractions.

Integration is an art. Take integration by parts, for example. How you decide to split an expression into u and dv makes all the difference in the world. One choice makes the problem easy to solve while other choices result in a more complex integral than the original problem.

The method to use is the one that works on the problem at hand. There are some general heuristics, but they remain heuristics.

I can usually tell whether someone has integrated some equation by hand or used a program such as Maple and Mathematica to do the job for them. The programs apply the heuristics and come up with a page-long equation. Done nicely by hand, the same integral is expressed one or two lines of math.

Basically, integration is a "box of magic tricks" - you listed 4 of the common ones. Unlike differentiation, it's not a logical process where you can follow a set of "rules" and guarantee to get the answer.

The only way to find out which trick to use for a given integral is by solving lots of problems. Try one trick, and if it doesn't seem to help then try another one.

You can use sites like that as a learning tool - look at the results (the simpler ones anyway!) and think how you could prove them using the integration methods you know. But DON'T try to learn all the answers!

You have the wrong primary focus on this.
Your primary focus when trying to integrate something should be:
Can I transform the integrand so that the result from the transformation is something I know how to integrate?

As long as the transformation itself is a permissible mathematical action, everything is allowed to be tried out.